The conic equation 16 · x² + 4 · y² + 96 · x - 8 · y + 84 = 0 is equivalent to (x + 3)² / 4.75 + (y - 2)² / 19 = 1, that represents an ellipse.
How to derive the nature of a conical by algebraic means
Mathematically speaking, conics are described by quadratic equations of the form A · x² + B · y² + C · x + D · y + E = 0, where A, B, C, D, E are real coefficients of the conic.
We can infer if a conic is a circle, a parabola, an ellipse or a hyperbola by modifying the expression into its standard form by algebra properties. First, write the equation of the conic:
16 · x² + 4 · y² + 96 · x - 8 · y + 84 = 0
Second, complete the square in the equation of the previous step:
16 · (x² + 6 · x) + 4 · (y² - 2 · y) = - 84
16 · (x² + 6 · x + 9) + 4 · (y² - 2 · y + 4) = - 84 + 16 · 9 + 4 · 4
16 · (x² + 6 · x + 9) + 4 · (y² - 2 · y + 4) = 76
Third, factor the polynomial and simplify the expression until standard form is derived:
16 · (x + 3)² + 4 · (y - 2)² = 76
(x + 3)² / 4.75 + (y - 2)² / 19 = 1
The conic equation represents an ellipse.
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Once an antibiotic is given, number of bacteria decreases at a rate of 15%/day. There were about 15,000 bacteria prior to the treatment. The graph models the number of bacteria after x days of treatment.
What are the key features of the exponential function modeled in terms of this situation and how can they be interpreted?
Select each correct answer.
The line y = 0 is an asymptote of the graph.
The x-intercept is 30.
The line x = 0 is an asymptote of the graph.
The asymptote indicates that there will never be more than 15,000 bacteria.
The y-intercept is 15,000.
The y-intercept represents the number of bacteria when treatment began.
The asymptote indicates that as time increases, the number of bacteria approaches 0.
The key features of the exponential function modeled in terms of this situation and how can they be interpreted are;
1) The y-intercept is 15,000.
2) The y-intercept represents the number of bacteria when treatment began.
3) The asymptote indicates that as time increases, the number of bacteria approaches 0.
What are the key features of the exponential function Graph?The general formula for an exponential function is;
y = aˣ
Now, an asymptote of a curve is defined as a line that exists in a way that the distance that exists between the curve and the line approaches zero as either one or both of the x or y coordinates approaches to infinity.
From the given graph, we can say that the y-intercept which is where the curve intersects the y-axis is 15,000 and this indicates the number of bacteria that existed prior to treatment.
The asymptote of the graph would likely be around x = 1 and it indicates that as time increases, the number of bacteria approaches 0.
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Solve each given equation and show your work. Tell whether it has one solution, an infinite number of solutions, or no solutions, and identify each equation as an identity, a contradiction, or neither.
A) 6x + 4x - 6 + 24 + 9x
B) 25 - 4x = 15 - 3x +10 - x
C) 4x + 8 = 2x + 7 + 2x - 20
(i) The value of x is equal to 30 and the equation has one solution and it is neither identity nor contradiction.
(ii) This equation has infinite number of solutions and it is an identity.
(iii) This equation has no solutions and it is a contradiction.
What is identity and contradiction?
An identity is an equation that can be solved using any integer that can replace the variable in the equation in a meaningful way. A conditional equation is one that is met by some numbers but not by others, such as 2x = 4. A contradiction is an equation, such as x = x + 1, that cannot be solved.
We know that,
An equation having infinitely many solutions is called an identity equation.
An equation that having no solution is called a contradiction.
(i) 6x + 4x - 6 + 24 + 9x
Combining like terms
x = 30
Therefore, The value of x is equal to 30 and the equation has one solution and it is neither identity nor contradiction.
(ii) 25 - 4x = 15 - 3x +10 - x
Combining like terms
0 = 0
It means, The equation is true for all values of x.
Therefore, This equation has infinite number of solutions and it is an identity.
(iii) 4x + 8 = 2x + 7 + 2x - 20
Combining like terms
8 = -13
It means, The equation is not true for any value of x.
Therefore, This equation has no solutions and it is a contradiction.
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A company hired 20 new graduates. The mean salary of the newly hired employees is 38 in thousands of dollars, where Σ(x - 38)² = 64. Which is the population variance of the given data ?
The population variance of the given data is 3.2.
Given Σ(x - 38)² = 64 .
We know that the population variance of the data is the square of the standard deviation.
Now the standard deviation for the population is given by
σ =√(Σ(x - 38)² / N)
Again variance = σ ²
Let us first calculate σ .
σ = √(64/20) = √3.2
variance
= σ ²
= (√3.2)²
= 3.2
Therefore the variance of the population is 3.2
In probability theory and statistics, variance is the predicted squared variation of a random variable from its population mean or sample mean. Dispersion, or how far apart from the mean a set of data are from one another, is measured by variance.
A few ideas that use variance include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling.
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If 2 is added to a number and the sum is tripled, the result is 16 more than the number. Find the number.
Answer:
5
Step-by-step explanation:
5 + 2 = 7
7(3) = 21
_
5 + 16 = 21
If 10 students from a 1st-grade class are lining up to enter the cafeteria line for lunch, how many ways are there to line up the 10 students?
Answer: The answer is 10
Step-by-step explanation:
What is 534.8 x 17 in long multiplication?
Answer:
Step-by-step explanation: 9091.6
Answer:
9091.6
Step-by-step explanation:
help plss
Write the description of each transformation in the order of operations.
Answer: PEMDAS
Parenthesis exponents multiplication Division addition subtraction
Step-by-step explanation:
solve (x+1) first then after just follow pemdas
SOMEONE PLEASE HELPPPPPPPPPPPP!
Given: (GH) is a midsegment of ∆DEF. Show all work.
Find the value of x and y.
What is the length of (DF)?
In the given ΔDEF x is 17 units and y is 2 units.
What is a triangle?A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.
Given, GH is the midpoint of ΔDEF and EF is the base of the ΔDEF with a length of 28 units.
We know the midpoint of ΔDEF is (1/2) EF.
∴ GH = (28/2) = 14 units.
Now GH = x - 3.
14 = x - 3.
x = 17.
Given DH and FH are equal.
∴ y + 8 = x - 7.
y + 8 = 17 - 7.
y + 8 = 10.
y = 2.
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**BRAINLEST**, **THANKS** AND **5 STARS** TO WHOEVER ANSWERS CORRECT FIRST! PLEASE HELP ME OUT THIS IS FOR ALGERBRA 2/PRE-CALCULUS
- During a certain time of year, the daily temperature in a certain city, in degrees Celsius follows a periodic pattern. The graph below shows the temperature over two days, where time t is measured in hours after 12:00 a.m. (midnight) on the first day.
- Write an equation for the graph below in terms of Y (temperature in degrees Celsius) and T (time in hours) to represent the given context.
- ANSWER SHOULD LOOK LIKE THIS FORMULA
y = ( a ) sin ( b ) (t - h) + k
- IF YOUR STILL UNCLEAR, THE SECOND SCREENSHOT IS WHAT THE ANSWER SHOULD LOOK LIKE
- (NO, THAT SECOND PHOTO WITH THE FORMULA ISNT THE ANSWER!!!!!)
GRAPH IS GIVEN BELOW PLEASE HELP ME OUT! ILL GIVE BRAINLIEST VOTE 5 STARS AND GIVE THANKS IF YOURE CORRECT AND FAST!!!! PLEASE HELP ME OUT
The sinusoidal equation for the graph in terms of Y, the temperature in degrees Celsius) and the time, T in hours is; Y = 3·sin((π/12)·(T + 6)) + 15
What is a sinusoidal equation?A sinusoidal equation or function is an oscillating and periodic function that is based on the sine of an angle.
The required form of the response is y = a·sin(b·(t - h)) + kThe function of the graph is a sinusoidal function with the parts;
a = The amplitude
b = The frequency = 1/T
T = The period of the function
k = The vertical shift of the function
h = The horizontal shift of the function
From the graph, we have;
The minimum point on the graph of the function = (1, 12)
The maximum point on the graph of the function = (13, 18)
The amplitude of the graph a is therefore; a = (18 - 12)÷2 = 3
The period of the graph = 1/b
1 cycle is completed in 24 hours, which indicates that the frequency of the function is; f = 2·π/24 = π/12 = b
The vertical shift, k from the parent function, y = sin(x) which is the distance the midline of the graph is raised above the x-axis is; k = 15
sin((π/12)·(0 - h)) = -1
((π/12)·(0 - h)) = -π/2
-h = 6
h = -6
The function, y = a·sin(b·(t - h)) + k, is therefore;
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Two identical rectangles linked at one corner are drawn on coordinate axes. Give the coordinates of point A and point B
The coordinates of the vertices of the points A and B are;
A(9, 7), B(15, 9)
What is the location of a point on the coordinate plane?A point on the coordinate plane is described by a pair of points known as an ordered pair.
The coordinates of the vertices of the identical rectangles are; (3, 5), (9, 5), (9, 9)
The length of (9 5) from A is the same as the length from (9, 9) to A.
The coordinate of the point A is therefore;
x-coordinate = 9
y-coordinate = 5 + (9 - 5)/2 = 7
The coordinate of point A is (9, 7)
The length of the sides of the rectangle are;
Length = 9 - 3 = 6
Width = 9 - 7 = 2
The coordinates of the point B are;
x-coordinate = 9 + 6 = 15
y-coordinate = 9
The coordinates of the point B is (15, 9)
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a =? , b = 40 ft , C = 50 ft
Answer:
a = 30
Step-by-step explanation:
I am not exactly sure what the question is asking but if referring to right triangles, the Pythagorean theorem would work best.
Formula :
[tex]a^2 + b^2 = c^2\\\\[/tex]
Substitute values :
[tex]a^2 + 40^2 = 50^2\\a^2 + 1600 = 2500[/tex]
Solve for a:
[tex]a = \sqrt{2500 - 1600} \\a = 30[/tex]
** you could also use the 3:4:5 ratio
I'm very stuck on this problem, mainly because of how restricted the selection of proofs is, and if you can help or give ideas I would really appreciate just listening to your thoughts on the question, thank you!
Check the picture below.
so in the parallelogram, the two triangles split by the side AC are congruents as you can see there by AAS, that means their sides and angles are congruent, that also includes the exterior angles.
since the exterior angles are also congruent, we can extend lines BC and AD as you see there in green and run an exterior angle at vertices A and C, since those two exterior angles are congruent and are also alternate interior ones, voila! it must be that BC || AD.
Write the equation in slope-intercept form.
3x + 6y = 18
y =
=x + [
Answer:
Slope intercept form is achieved by isolating y to one side. 3x-6y=18: given. -6y=-3x+18: algebra. y=3/6x-3: simplifying.
1.3 x 10 to the power of negative 3
Answer:0.0013
Step-by-step explanation:
Answer: the answer is on the calclulator
Step-by-step explanation:
Solve one sixth times quantity x plus 24 end quantity equals negative 6
The value of one-sixth times quantity x plus 24 end quantity equals negative 6 is -180.
What is an expression?The expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.
In this case, the expression given will be illustrated as:
1/6x + 24 = -6
Collect like terms
x / 6 = -6 - 24
x/6 = -30
Cross multiply
x = -30 × 6
x = -180
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A construction worker earns an average bi-weekly net pay of $1,764.00. Which compound inequality correctly shows the amount of money, t, the worker can spend if the monthly budget for food is between 10% and 20%?
$176.40 ≤ t ≤ $352.80
$352.80 ≤ t ≤ $706.50
$382.20 ≤ t ≤ $764.40
$420.42 ≤ t ≤ $573.30
The compound inequality is $176.40 ≤ t ≤ $352.80 if a construction worker earns an average bi-weekly net pay of $1,764.00 option (A) Is correct.
What is inequality?It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.
It is given that:
A construction worker earns an average bi-weekly net pay of $1,764.00.
= 10% of 1764
= (10/100)1764
= $176.4
= 20% of 1764
= (20/100)1764
= $352.8
Inequality will be: $176.40 ≤ t ≤ $352.80
Thus, the compound inequality is $176.40 ≤ t ≤ $352.80 if a construction worker earns an average bi-weekly net pay of $1,764.00 option (A) Is correct.
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Answer:
$382.20 ≤ t ≤ $764.40
Step-by-step explanation:
Hello, so first we would need to convert bi-weekly pay to monthly income.
First we would times $1,764.00 by 26
1764.00 × 26 = 45864
Then we would divide 45864 by 12
45864 ÷ 12 = 3822.
Then we calculate how much the worker can spend if their monthly budget for food is in between 10% and 20%
3822 × 10% = 382.20
3822 × 20% = 764.40
So, we can see the correct answer is $382.20 ≤ t ≤ $764.40.
Simplify:
a+b
a-b
[?]
a=5 and b=3
Simplification of the given expression ( a + b ) / ( a - b ) for the given values of a and b is equal to ( 4 / 1 ).
As given in the question,
Given expression:
( a + b ) / ( a - b )
Simplification of the given expression ( a + b ) / ( a - b ) for the value
a = 5 and b = 3 is given by,
( a + b ) / ( a - b ) we get,
( a + b ) / ( a - b )
= ( 5 + 3 ) / ( 5 - 3)
= ( 8 ) / ( 2 )
= ( 4 × 2 ) / ( 2 )
= ( 4 / 1 )
Therefore, simplification of the given expression ( a + b ) / ( a - b ) for the given values of a and b is equal to ( 4 / 1 ).
The complete question is:
Simplify the given expression:
(a + b)/ (a -b) = ?
For a=5 and b=3
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Mrs.Josey and her family were taking a trip to Dallas to go to the Botanical Gardens.If the trip was a total of 350 miles one way and their car can go 20 miles on one gallon of gas, how many gallons will they need to make it to Dallas
The number of gallons of gas which Mrs. Josey and her family would need is; 17.5 gallons of gas.
How much gas is needed to make it to Dallas?It follows from the task content that the quantity of gas in gallons needed by Mrs. Josey and her family to make it to Dallas is to be determined.
Therefore, since one gallon of gas is enough for 20 miles, It follows that the amount of gas required for 350 miles can be determined by proportion as follows;
20/1 = 350/x
x = 350 / 20
x = 17.5 gallons of gas.
Ultimately, the amount of gas needed for Mrs. Josey and her family to make it to Dallas is; 17.5 gallons of gas.
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The figure shows right triangle MPR on the coordinate plane. Line segment MQ is
a median of the triangle.
What are the coordinates of point Q? Explain your answer. Using the information in the figure, show that the length of the median MQ is half the length of the hypotenuse of MRP. Provide an argument supported by valid mathematical
reasoning and/or calculations.
Answer:
see explanation
Step-by-step explanation:
the median is a segment that goes from the vertex of a triangle to the midpoint of the opposite side.
Then Q is the midpoint of PR
given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint M is
M = ( [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] ) ← midpoint formula
use this formula to find the coordinates of Q
with (x₁, y₁ ) = R (0, 2b ) and (x₂, y₂ ) = P (2a, 0 )
Q = ( [tex]\frac{0+2a}{2}[/tex] , [tex]\frac{2b+0}{2}[/tex] ) = ( [tex]\frac{2a}{2}[/tex] , [tex]\frac{2b}{2}[/tex] ) = (a, b )
---------------------------------------------------
use the distance formula to find the lengths of median MQ and hypotenuse PR of Δ MRP
d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = M (0, 0 ) and (x₂, y₂ ) = Q (a, b )
MQ = [tex]\sqrt{(a-0)^2+(b-0)^2}[/tex] = [tex]\sqrt{a^2+b^2}[/tex]
repeat with (x₁, y₁ ) = P (2a, 0 ) and (x₂, y₂ ) = R (0, 2b )
PR = [tex]\sqrt{(0-2a)^2+(2b-0)^2}[/tex]
= [tex]\sqrt{(-2a)^2+(2b)^2}[/tex]
= [tex]\sqrt{4a^2+4b^2}[/tex]
= [tex]\sqrt{4(a^2+b^2)}[/tex]
= 2[tex]\sqrt{a^2+b^2}[/tex]
then MQ is half PR
use the drop down menus for 5−4+7+1
Answer:
9
Step-by-step explanation:
5 - 4 + 7 + 1 = 9/1 = 9
A spice mixture is 25% thyme. How many grams of thyme must be added to 12g of the mixture to increase the thyme content to 40%?
Answer:
3 g
Step-by-step explanation:
You want to know the mass of thyme that must be added to 12 g of a spice mix that is 25% thyme in order to make it 40% thyme.
SetupLet t represent the amount of added thyme (in grams). The amount of thyme in the augmented mix is ...
0.25(12) + t = 0.40(12 +t)
SolutionSimplifying the equation gives ...
3 +t = 4.8 +0.40t
0.6t = 1.8 . . . . . . . . . subtract 0.40t +3
t = 3 . . . . . . . . . . . divide by 0.6
3 grams of thyme must be added to the mix.
somebody please help me
Answer:
I got (-1,-4) and (3,0) When I graphed with Desmos. I do no see that solution.
Step-by-step explanation:
The volume of 10,000 drops of a liquid is 10 fluid ounces.
What is the volume of 10 drops?
Answer: .01 fluid ounces.
Step-by-step explanation:
1,000 drops is 1 FL
100 drops is .1 FL
10 drops is .01 FL
help on this maths
im confuse and dont know what to do
Answer:
30-2=28+542=570
Step-by-step explanation:
A bell rings every 1/6 hour. Assuming it just rang, how many times will it ring in the next 2/3 hour?
Thomas has $10 to spend on lunch. He decides to buy come chicken sandwiches, which cost $2 each, and one container of orange juice, which cost $1.50. What is the greatest number of chicken sandwiches that Thomas can buy?
1. A candle weighs 300 grams (g). Each hour the candle is lit, it burns 4 g of its wax. a. Leth be the number of hours the candle burns. Write an equation that can be used to determine the number of hours it takes for the candle to weigh 206.4 g. b. Solve the equation to determine the number of hours it takes for the candle to weigh 206.4 g. Show your work.
Please help me tomorrow I Will hace a test about this question!!
Answer:
Step-by-step explanation:
300 grams-24 hour= 276 so i think u just need to keep doing this till it gets to 206.4
The equation which represents the number of hours will be ln(206.4) = 1200ln(0.014).
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that a candle weighs 300 grams (g). Each hour the candle is lit, it burns 4 g of its wax. a. Leth be the number of hours the candle burns.
The equation for the number of hours will be calculated as,
[tex]h=300\times (0.014)^h[/tex]
lnh= 300 x ln(0.014)
Therefore, the equation which represents the number of hours will be ln(206.4) = 1200ln(0.014).
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solve by factoring 6^2 − x - 12 = 0
[tex]{ \purple{ \tt{x = 3}}}[/tex]
[tex]{ \blue{ \sf{ {6}^{2} - x - 12 = 0}}}[/tex]
[tex]{ \blue{ \sf{36 - x - 12 = 0}}}[/tex]
Take 12 as common
[tex]{ \blue{ \sf{ \cancel{12}(3 - x) { \cancel{-12} = 0}}}}[/tex]
[tex]{ \blue{ \sf{3 - x = 0}}}[/tex]
[tex]{ \red{ \boxed{ \green{ \tt \overbrace{ \underbrace{x = 3}}}}}}[/tex]
Divide:
4⁄6 ÷ 2⁄6 =
..
Answer:2
Step-by-step explanation: I don’t have a explanation
Answer: 2
Step-by-step explanation:
You should be able to do this...
how to solve a linear equation if x and y is zero
Answer:
0
Step-by-step explanation:
if we x and y = 0
if we have 3x + 3y= 0